Precalculus Examples

$f (x) = x^{4} - x^{5}$

Step 1

Set $x^{4} - x^{5}$ equal to $0$ .

$x^{4} - x^{5} = 0$

Step 2

Solve for $x$ .

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Step 2.1

Factor $x^{4}$ out of $x^{4} - x^{5}$ .

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Step 2.1.1

Multiply by $1$ .

$x^{4} \cdot 1 - x^{5} = 0$

Step 2.1.2

Factor $x^{4}$ out of $- x^{5}$ .

$x^{4} \cdot 1 + x^{4} (- x) = 0$

Step 2.1.3

Factor $x^{4}$ out of $x^{4} \cdot 1 + x^{4} (- x)$ .

$x^{4} (1 - x) = 0$

Step 2.2

If any individual factor on the left side of the equation is equal to $0$ , the entire expression will be equal to $0$ .

$x^{4} = 0$

$1 - x = 0$

Step 2.3

Set $x^{4}$ equal to $0$ and solve for $x$ .

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Step 2.3.1

Set $x^{4}$ equal to $0$ .

$x^{4} = 0$

Step 2.3.2

Solve $x^{4} = 0$ for $x$ .

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Step 2.3.2.1

Take the specified root of both sides of the equation to eliminate the exponent on the left side.

$x = \pm \sqrt[4]{0}$

Step 2.3.2.2

Simplify $\pm \sqrt[4]{0}$ .

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Step 2.3.2.2.1

Rewrite $0$ as $0^{4}$ .

$x = \pm \sqrt[4]{0^{4}}$

Step 2.3.2.2.2

Pull terms out from under the radical, assuming positive real numbers.

$x = \pm 0$

Step 2.3.2.2.3

Plus or minus $0$ is $0$ .

$x = 0$

Step 2.4

Set $1 - x$ equal to $0$ and solve for $x$ .

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Step 2.4.1

Set $1 - x$ equal to $0$ .

$1 - x = 0$

Step 2.4.2

Solve $1 - x = 0$ for $x$ .

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Step 2.4.2.1

Subtract $1$ from both sides of the equation.

$- x = - 1$

Step 2.4.2.2

Divide each term in $- x = - 1$ by $- 1$ and simplify.

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Step 2.4.2.2.1

Divide each term in $- x = - 1$ by $- 1$ .

$\frac{- x}{- 1} = \frac{- 1}{- 1}$

Step 2.4.2.2.2

Simplify the left side.

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Step 2.4.2.2.2.1

Dividing two negative values results in a positive value.

$\frac{x}{1} = \frac{- 1}{- 1}$

Step 2.4.2.2.2.2

Divide $x$ by $1$ .

$x = \frac{- 1}{- 1}$

Step 2.4.2.2.3

Simplify the right side.

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Step 2.4.2.2.3.1

Divide $- 1$ by $- 1$ .

$x = 1$

Step 2.5

The final solution is all the values that make $x^{4} (1 - x) = 0$ true.

$x = 0, 1$

Step 3